A self-adjoint operator is a type of linear operator in mathematics, particularly in the field of functional analysis. It is defined on a Hilbert space and has the property that it is equal to its own adjoint. This means that for any two vectors in the space, the inner product of the operator applied to one vector with the other is the same as the inner product of the first vector with the operator applied to the second. Self-adjoint operators are important because they have real eigenvalues and their eigenvectors corresponding to different eigenvalues are orthogonal. This property makes them useful in various applications, including quantum mechanics, where self-adjoint operators represent observable quantities. Examples of self-adjoint operators include Hermitian matrices and differential operators under certain conditions.